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The performance of Orthogonal Matching Pursuit (OMP) for variable selectionis analyzed for random designs. When contrasted with the deterministic case,since the performance is here measured after averaging over the distribution ofthe design matrix, one can have far less stringent sparsity constraints on thecoefficient vector. We demonstrate that for exact sparse vectors, theperformance of the OMP is similar to known results on the Lasso algorithm[\textit{IEEE Trans. Inform. Theory} \textbf{55} (2009) 2183--2202]. Moreover,variable selection under a more relaxed sparsity assumption on the coefficientvector, whereby one has only control on the $\ell_1$ norm of the smallercoefficients, is also analyzed. As a consequence of these results, we also showthat the coefficient estimate satisfies strong oracle type inequalities.

Variable Selection in High Dimensions with Random Designs and Orthogonal Matching Pursuit